pkg load symbolic

syms y(t) t ;
f=@(t,y) t-y ;
egyenlet=diff(y(t),t)==t-y ;
sol=dsolve(egyenlet,y(0)==1) ;
fsol=matlabFunction(rhs(sol)) ;
tb=0:0.5:3;
yb=-1:0.5:5;
[T,Y]=meshgrid(tb,yb);
dY=f(T,Y);
dT=ones(size(dY));
quiver(T,Y,dT,dY);
hold on;
xx=linspace(tb(1),tb(end));
plot(xx,fsol(xx));
axis([tb(1),tb(end),yb(1),yb(end)])

warning: worked around octave bug #42735
warning: called from
    minus at line 42 column 5

fixpont-iteráció¶

y^{'} = f (t, y) y (a) = b

$y'=f(t,y)\ \ y(a)=b$

y (t) = b + \int_{a}^{t} f (s, y (s)) d s

$y(t)=b+\int_a^t f(s,y(s))ds$

y \to T (y), ahol T (y) (t) = b + \int_{a}^{t} f (s, y (s)) d s

$y \to T(y), \ \text{ahol }\ T(y)(t)=b+\int_a^t f(s,y(s))ds$

y_{n} \to y_{n + 1} = T (y_{n}), ahol T (y_{n}) (t) = b + \int_{a}^{t} f (s, y_{n} (s)) d s

$y_{n} \to y_{n+1}=T(y_n), \ \text{ahol }\ T(y_n)(t)=b+\int_a^t f(s,y_n(s))ds$

f (t, y) = t - y, y_{0} (t) = 1

$f(t,y)=t-y,\ \ y_0(t)=1$

y_{1} (t) = 1 + \int_{0}^{t} (s - 1) d s = 1 - t + \frac{t^{2}}{2}

$y_1(t)=1+\int_0^t (s-1) ds= 1-t+\frac{t^2}{2}$

y_{2} (t) = 1 + \int_{0}^{t} (s - 1 - \frac{s^{2}}{2} + s) d s = 1 - t + t^{2} - \frac{t^{3}}{6}

$y_2(t)=1+\int_0^t (s-1-\frac{s^2}{2}+s) ds= 1-t+t^2-\frac{t^3}{6}$

y_{3} (t) = 1 + \int_{0}^{t} (s - 1 + s - s^{2} + \frac{s^{3}}{6}) d s = 1 - t + t^{2} - \frac{t^{3}}{3} + \frac{t^{4}}{24}

$y_3(t)=1+\int_0^t (s-1+s-s^2+\frac{s^3}{6}) ds= 1-t+t^2-\frac{t^3}{3}+\frac{t^4}{24}$

y_{4} (t) = 1 + \int_{0}^{t} (s - 1 + s - s^{2} + \frac{s^{3}}{3} - \frac{t^{4}}{24}) d s = 1 - t + t^{2} - \frac{t^{3}}{3} + \frac{t^{4}}{12} - \frac{t^{5}}{120}

$y_4(t)=1+\int_0^t (s-1+s-s^2+\frac{s^3}{3}-\frac{t^4}{24}) ds= 1-t+t^2-\frac{t^3}{3}+\frac{t^4}{12}-\frac{t^5}{120}$

quiver(T,Y,dT,dY);
hold on;
xx=linspace(tb(1),tb(end));
plot(xx,fsol(xx),'.-');
plot([tb(1),tb(end)],[1,1]);
uj=@(t) 1-t+1/2*t.^2;
plot(xx,uj(xx),';1;');
uj=@(t) 1-t+t.^2-1/6*t.^3;
plot(xx,uj(xx),';2;');
uj=@(t) 1-t+t.^2-1/3*t.^3+1/24*t.^4;
plot(xx,uj(xx),';3;');
uj=@(t) 1-t+t.^2-1/3*t.^3+1/12*t.^4-1/120*t.^5;
plot(xx,uj(xx),';4;');

axis([tb(1),tb(end),yb(1),yb(end)])